Study of symmetry in Constraint Satisfaction Problems? Belaid Benhamou. URA CNRS Universite de Provence,

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1 Study of symmetry in Constraint Satisfaction Problems? Belaid Benhamou URA CNRS Universite de Provence,,Place Victor Hugo - F11 Marseille cedex, France phone number : Benhamou@gyptis.univ-mrs.fr Abstract. Constraint satisfaction problems (CSP's) involve nding values for variables subject to constraints on which combinations of values are permitted. Symmetrical values of a CSP variable are in a sense redundant. Their removal will simplify the problem space. In this paper we give the principle of symmetry and show that the concept of interchangeability introduced by Freuder, is a particular case of symmetry. Some symmetries can be computed eciently thanks to the structure of the problem (neighborhood interchangeability is a kind of these symmetries). Therefore we show how such symmetries can be used by existing constraint propagation algorithms and introduce a backtrack procedure exploiting symmetries. Both theoritical analysis and expiriments indicate that our proposed approach is an improvment of neighborhood interchangeability use, and has very good behavior for pigeon-hole problems. 1 Introduction The nite domain constraint satisfaction problem (CSP) 2 is well known in Articial Intelligence. It has been investigated in the 2 Through out this paper, we use CSP to refer to the nite domain constraint satisfaction problem. past by a number of researchers in dierent contexts; and steal a well-studied research area of recent years (refer to Kumar [1]). A CSP involves, (1) a (nite) set V = fv 1 ; v 2 ;... ; v n g of variables, (2) a - nite set D = fd 1 ; D 2 ;... ; D n g of discret domain values in which D i is the nite discrete domain associated whith the variable v i ; to avoid confusions between values of dierent domains, d i will denote the fact that it belonges to the domain D i, () a nite set C = fc 1 ; c 2 ;... ; c n g of constraints, a k-ary constraint c i is dened on a subset V k V of variables which we denote var(c i ), (4) and a nite set R = fr 1 ; R 2 ;... ; R n g of relations corresponding to the constants in C, R i represents the list of tuples form in which the tuples of values satisfying the constraint c i are enumerated. Thus, a CSP can be seen as a quadriplet P(V; D; C; R). A value assignment is a mapping which species a value for each variable: formally a value assignment I can be seen as: I : V! [ i2[1;n]d i such that I[v i ] 2 D i, 8i 2 [1; n]. A value assignment satises a constraint if it gives a combination of values to variables that is permitted by the constraint; otherwise it falsies it. Thus a constraint satisfaction problem is the task of nding one or all value assignments for the constraints network such that all the constraints are satised together. As beeing expected, various techniques for solving CSP's have been developed; these include backtraking, arc consistancy (Waltz [1], Mackworth [11]), path consistan-

2 cy (Freuder [6] On other hand, symmetries for boolean constraints are well studied in (Benhamou and Sais [2,]). They showed that it is a real improvment for eciency of several automated deduction algorithms. In this paper we develop the concept of symmetry for CSP's. Symmetrical domain values will be in a sense redundant. Their removal will simplify the problem search space. On other hand the set of solution of a CSP can be represented in a more compact way using symmetry. Indeed only non-symmetrical solutions are computed (basical solutions) from which we process the other solutions whitout duplication of eorts. The paper is organized as following : Two levels of semantic symmetry are de- ned in Section 2. Section discusses syntactical symmetry which is a form of semantic symmetry that can be computed eciently using only the structure of the considered problem. In other words, syntactical symmetry is considered as a susient condition to hold semantic symmetry (Neighborhood interchangeability (Freuder [7]) is a case of syntactical symmetry). Section 4 explains how symmetrical values can be used in various algorithms such as propagation methods and propose a backtrack procedure taking advantage of symmetrical values. In section 5 we evaluate the proposed techniques by experimental results. Section 6 concludes the work. For simplicity we studie binary CSP's, which involve only constraints between two variables. However, symmetry remains available for non-binary CSP's; and non-binary CSP's can be transformed into binary ones (Rossi, Dhar and Petri [12]). 2 Semantic symmetry We are interested by two problems in CSP's : the problem of nding a solution (test of satisability) and the problem of ndind all the solutions of the CSP. Thus two levels of semantic symmetry are dined whith respect to the two previous problems. Denition 1 Symmetry for satisability. Two domain values b i and c i for a CSP variable v i 2 V are symmetrical for satisability (notation b i c i ) i the following assertions are equivalent : 1. There is a solution of the CSP which contains the value b i ; 2. There is a solution of the CSP which contains the value c i. Domain values can be not only symmetrical for satisability (denition 1) but symmetrical for the set of all solutions as well. Thus, if sol(p) denotes the set of solutions of the CSP P, then we dene a second level of semantic symmetry as follow : Denition 2 Symmetry for all solutions. Two domain values b i and c i for a CSP variable v i 2 V are symmetrical for sol(p) (notation b i ' c i ) if and only if each solution of the CSP containing the value b i can be mapped into a solution containing the value c i and viceversa. Remark. Symmetrical values for all solutions (denition 2) are also symmetrical values for satisability (denition 1). Example 1 Graph coloring problem. The problem consists in coloring the vertices so that no two vertices which are joined by an edge have the same color. The available colors (domain values) at each vertex are shown (gure 1). The red 1 and white 1 colors for vertex v 1 are two symmetrical domain values. Indeed, solutions in which one of them participates, can

3 be obtained from the solutions in which the other value appears by permuting the values red and white for the variables v 1, v 2 and v. sucient condition to hold semantic symmetry (denition 2) and give an ecient method for search of such symmetries. v1 red, white c1 Syntactical symmetry marroon, red, white v Identiying semantic symmetries as dined in c12 c4 (denitions 1 and 2) is straightforward time consoming, as this requires solving the problem. This section studies a family of symmetries (syntactical symmetries) which are v2 blue, red, white blue v4 c24 more tractable computationally, thanks to the structures of the considered problem. Fig. 1. The graph coloring problem. A permutation of domain values of a binary CSP P = (V; D; C; R) can be seen In other hand, Freuder introduced in ([7]) as: : [ i2[1;n]d i! [ i2[1;n]d i, such that (d i ) 2 D i, 8i 2 [1; n] and 8d i 2 D i. The the notion of interchangeability, where two domain values are interchangeable in some en- fv; D; Cg of the CSP P. However, it induces permutation have no inuence on the sets veronment, if they can be substituted for each a permutation t on the tuples in each relation other without any eects to the environment. R ij 2 R and then a permutation R on the Let us summarize the main denition. relations themselves. Therefore a syntactical symmetry of a CSP P = (V; D; C; R) is a permutation of domain values which leaves the CSP P invariant (i.e. R (R i ) = R i ; 8R i 2 R). Formally: Denition. Two domain values b i and c i for a CSP variable v i 2 V are fully interchangeable i (1) every solution to the CSP which contains b i remains a solution when c i is substituted for b i, (2) every solution to the C- SP which contains c i remains a solution when b i is substituted for c i. Remark. Interchangeable values are particular symmetrical values for all solutions in which the mapping consists to permute the interchangeable values and still identity for the other values. In the previous examples, values red 1 and white 1 are not interchangeable. Thus, the principle of symmetry seems to be more general than the notion of interchangeability. Therefore, eliminating symmetrical values can prune more great deal of eort from a backtrack search tree if such values are processed eciently. We study in the next section syntactical symmetry of domain values which is a Denition 4 Syntactical symmetry. A permutation is a syntactical symmetry of the CSP P = (V; D; C; R) i [8R ij 2 R, d i ; d j 2 tuples(r ij ) ) (d i ); (d j ) 2 tuples(r ij )]. Remark. A syntactical symmetry of a CSP is a domain value permutation such that R (R i ) = R i, 8R i 2 R. Example 2 Pigeon-hole problem. The problem consists in putting n pigeons in n1 holes such that each hole holds at most one pigeon. Take for instance 4 pigeons and holes. The pigeons are represented by the set of variables the holes by the domain values, as it was shown Both t resp. R are natural generalizations for to tuples resp. relations.

4 in the constraint graph of gure 2, the constraint c 1 is given in its microstructure form showing the permitted tuples in the relation R 1. v2 c12 a b c24 c14 v1 a b c J@ L@ AS JJ LL@@ AA SS v4 c2 a b c S JJ SS c4 c1 c b a Fig. 2. Pigeon-hole problem for 4 pigeons and holes. The permutation dened as: (a i ) = (b i ), (b i ) = (c i ), (c i ) = (a i ), 8i 2 [1; 4] keeps the CSP invariant (i.e. R (R i ) = R i, 8i 2 [1; 4]). Thus, it is a symmetry of the CSP. Denition 5. Two domain values b i and c i for a CSP variable v i 2 V are syntactically symmetrical (notation b i c i ) if there exists a syntactical symmetry of the CSP P such that (b i ) = c i. Remark. The relation () is a relation of e- quivalence. In the previous example domain values a 1 and b 1 of the variable v 1 are syntactically symmetrical. Denition 6. A set fa 1 ; ;... ; 1 a2 an g of domain values form a cycle of symmetry in 1 1 P, if there exists a syntactical symmetry of P such that (a 1 1) = a 2 ; 1 1) = (a2 a ;... ; ) = a n ; 1 (an1 (an 1 1 1) = a 1 1 v Example. The sets of values fa i ; b i ; c i g, i 2 [1; 4] of the previous example forme four cycles of symmetry. All values in a cycle of symmetry are symmetrical two by two. Therefore, our method of search of symmetry will process a symmetry which gives for each domain, classes (cl(d i ) denotes the class of d i ) of values which are symmetrical together. Each classe will be identi- ed by a cycle of symmetry. Before, describing the search method of symmetry, we will prove that syntactical symmetry is a sucient condition for semantic symmetry. Theorem 7. If b i and c i are two syntactical symmetrical values of a CSP variable v i 2 V (b i c i ) then b i and c i are semantic symmetrical values for all solutions of the CSP (b i ' c i ). Proof. Cf. ([1]). Remark. Syntactical symmetrical values are also semantic symmetrical values. Symmetry expresses an important property that we use to make prune the search tree. Indeed if d i participates in no soliution of the CSP P and d i d i, then d i will participate in no solution too. Thus, we prune the sub-tree which corresponds to its assignment. Therefore, if there are n symmetrical domain values in cl(d), then we can cut n1 branches in the search tree if one of the domain values has already been identied that it paticipates in no solutions. See that neighborhood interchangeability is a very particular syntactical symmetry which permuts the interchangeable values and still identity for the other values. Such symmetries can not exists frequentlly. Our approache is more general and will get more use. Bellow we give the search method for syntactical symmetry.

5 .1 Search method for symmetry To be syntactically symmetrical, values need to satisfy some necessary conditions: Proposition 8. Let Rij (d i ) be the number of occurences of the value d i 2 D i in the relation R ij and tuples(r di ) ij the set of tuples of R ij in which d i appears, then to be syntactically symmetrical, values b i and c i must satisfy the following conditions: 1. Rij (b i ) = Rij (c i ), 8R ij 2 R; 2. for each d j 2 tuples(r bi ij ), 9 d j 2 tuples(r ci ij ) such that Rij (d j ) = Rij ( d j ), 8R ij 2 R. Proof. Cf. ([1]). The search method consits in two steps: (I) to partition each domain w.r.t the previous necessary conditions into primary classes of values which will be condidates for symmetry. (II) process a permutation from the primary classes which keeps the CSP invariant. We develope the step (II) which will give the complexity of the search method. procedure symmetry(d i 2 D) Repeat for each R ij 2 R: Repeat for each d i 2 D i, such that d i; d j 2 tuples(r ij): choose (d i) 2 cl(d i) and (d j) 2 cl(d j), such that (d i); (d j) 2 tuples(r ij) Fig.. The search symmetry algorithm The classes of symmetrical values are the dierent sycles of. A complexity bound for this algorithm can be found by assigning a worst case bound to each repeat loop. Given m relations, at most a values in each domain variable, we have the bound (the factors correspond to the repeat loops and the choose operation in topdown order): O(m a a 2 ) = O(m:a ). Bellow we show how several methods can be augmented with the advantage of symmetry. 4 Adaptation of various Constraints Propagation Algorithms Now we are in the position to show how these domain symmetrical values can be used to increase eciency of various existing algorithms. We give a few modications of the key procedures and show the advantages of the use of symmetry techniques for certain problem types. We focus on binary CSPs. 4.1 Constraint ltering algorithms The critical and most time consuming task in network consistency procedures is to check if all values of a particular variable domain can potentially be a member of a solution. These checks are done repetitively for singular variables w.r.t singular constrains. In the case of binary constraints, the procedure revise(d i,d j ) is usually used. It removes all values of D i for which no value of the domain D j can be found such that the binary constraint c ij between variables v i and v j is satised. It is abvious that the worst-case complexity of revise is O(a 2 ) where a is the maximum domain size. The procedure revise is applied on dierent constraints seperatelly, then symmetrical domain values must be computed w:r:t a given constraint c. The main idea is that domain values can be symmetrical w:r:t a constraint c, but not symmetrical w:r:t other constraints. So it is important to caracterize symmetrical values for each constraint of the network independently. We use the expression cl(d) v c (d is a domain value of the CSP variable v 2 var(c)) to denote the equivalence class of symmetrical values w:r:t to the constraint c in which d appears; formally: cl(d) v c =fd 2 D v : d dg. Figure 4 shows the procedure revise augmented by the advantage of symmetry.

6 procedure revise SV (var D i:domain,d j:domain) begin 4 i = fg repeat x := an element of D i 4 j = D j repeat y := an element of 4 j if x; y 2 tuples(r ij) then begin end 4 i := 4 i S fcl(x) v i c ij 4 j =f g end else 4 j := 4 j-fcl(y) v j until(4 j =f g) D i := D i fcl(x) v i until (D i =f g) D i := 4 i Fig.4. The revise SV c ij g algorithm. c ij g \ D ig The main dierence between the classical revise and revise SV is that the former checks in the worst case all tuples D i 2 D j and the later treats groups of symmetrical values e- qually. A symmetry dened on a constraint c partionnes the domain D i into subsets of domain values which are symmetrical together. If 5 v c is the set of symmetrical subsets domain values of the variable v 2 var(c), w:r:t the constraint c, if we assume that the sets 5c v for all constraint c and all variables v 2 var(c) are of size a (1 a a), then a worst case bound of the algorithm revise SV is O(a 2 ). 4.2 Backtrack search In the following, we want to envolve a tree search scheme where symmetrical search branches are recognized by use of symmetrical values. The algorithm is basically the same as classical backtrack tree search as discribed, for instance in ( Fox and Nadel [5]). But rst we have to give some notations we need for the development of the search procedure. Each output of a traditional backtrack procedure is an assignment tuple representing a solution for the given CSP. Because we want to handle groups of symmetrical values, we have to modify the form of the output. Instead of single assignment values, sets are used. As it was done in ([9]), assignment tuples are chifted to assignment bundles. Denition 9 Assignment Bundle. Let V be the set of n variables of the CSP P. An n-tuple 4 where the ith element (1 i n) is a non-vacuous subset of the domain D i is called an assignment bundle. Denition 1 Solution Bundle. Let sol(p) be the set of all solution of the CSP P. An assignment bundle 4 = f4 1 ;... ; 4 n g on the variables V of the CSP is said to be a solution bundle, if and only if n sol(p). Solutions bundle represent then groups of paths throught the search tree, which are solutions of the CSP. The terms of local and global consistency (see, for instance Dechter[4]) can be extended to assignment bundles. Denition 11. { An assignment bundle 4 p on the variables V p V is said to be locally consistent, if every assignment tuple extractable from 4 p is locally 4, consitent; { An assignment bundle 4 p on the variable V p V is said to be globally consistent, if there exists an extention assignment bundle 4 e on the variables (V V p ) such that 4 p [ 4 e is a solution bundle; { An assignment bundle 4 p is said to be inconsistent, if every assignment tuple extractable from 4 p is inconsistent (i.e., no tuple in 4 p can be extended to a solution). Now we modify the classical backtrack search such that for each pass a bundle assignment is computed. Solutions bundle regroup 4 I.e., all the constraints of the subnetwork de- ned by the variables V p are satised.

7 sets of symmetrical solutions. The following theorem gives the fundamental basis for the utilisation of symmetrical values. Theorem 12. Let 4 p be an assignment bundle on the variables V P V which is either globally consistent or inconsistent. Let v be a variable of V V p, v D v and C k all binary constraints from v to variables of (V V p ), such that the two following two conditions hold : will test both interchangeability and symmetry and compare them on two kind of problems: (I) randomlly generated CSP's, we use the same test model as proposed in Freuder ([8]). (II) the pigeon-hole problem which is known to be hard, is solved using symmetries with a linear complexity, however interchangeability get no use for this problem p + v is locally consistent; 2. 8d 1 ; d 2 2 v : 8c 2 C k ; d 1 d 2. Then 4 p + v is either globally consistent or inconsistent. Proof. Cf. ([1]) Random CSP's are characterized by the following four parameters: (1) n, the number of procedure backtrack SV (k:integer,b:assign-bundle); variables. (2) a, the maximum domain size. () begin t, the constraint tightness which is the fraction revise SV (D k; D p), for 1 p < k; of forbidden tuples to the number of possible for do some kind of look ahead ltringg d k := D k; repeat tuples. (4) the constraint density which is a number between and 1 given by d, indicates the fraction of additional constraints. x := an element of D k; C f := all constraints on v k to future variables; B[k] := ( cl(x) v k c2c f c ) \ d k; d k := d k B[k] if k = n then write(b) else backtracking SV (k + 1; B); until (d k = fg); 5.2 Results end. Fig.5. The backtrack algorithm. The advantageous behavior of the procedure backtracking SV is that symmetrical search branches are bundled and visited once. If a dead-end occurs, all the partial assignment extractable frome the derived assignment bundle are proven to be coniting. 5 experiments Now we want to invistigate the indicated performance improvement of our augmented search technique by experimental analysis. We 5.1 The experiment model Three forward-checking search procedures are compared: (1) (F C), the classical forwardchecking. (2) (F C N I), forward-checking with the advantage of neighborhood interchangeability seen as particular syntactical symmetry. () (F C SV ), the instance of the search scheme backtracking SV (see, gure 5) where forward-cheking ltring is used. The indicator of the complexity is the number of checks. Of cours, the checks needed for the computation of neighborhood interchangeability resp. symmetry are added to the run time checks. The samples of each test are randomly generated CSP's.

8 14 12 FC 1 FC-NI 8 FC-SV FC-SV Fig.6.Symmetry eects w.r.t the number of variables It can be seen in gure 6 that the eect of both symmetry and interchangeability grows if the problem increase. The variable size steps from 6 to 1, a is xed on 5, t and d are from the interval [.1-.4] (the protable ranges for the use of interchangeability as claimed in [8]). It can also be seen that F C SV denitily beats F C N I at these problems type FC FC-NI FC-SV Fig.7. FC and FC-NI eects on Pigeon-hole problem. Figure 8 shows that the complexity of F C SV for pigeon-hole problem seems to be linear, whereas both F C and F C N I (gure 7) cannot solve the problem when the number of pigeons is greater than eight, their complexities become quickly exponentiel when the number of pigeons is greater then 8. Thus, neighborhood interchangeability get no use for this problem. Fig.8. Symmetry eects on pigeon-hole problems. 6 Conclusion We have developed the formal cocept of symmetry in constraint satisfaction problems, then various constraintes satisfaction algorithmes can be adapted to exploit such information. The principe of interchangebility is shown to be a particular case of symmetry. Further investigation will consist to extend symmetry to domain values of dierent variables and try to identify certain type of C- SP's for which such symmetries get more use. References 1. B. Benhamou. Study of symmetry in constraint satisfaction problems. Technical Report 1, Universite de provence, B. Benhamou and L. Sais. Tractability through symmetries in propositional calculus. Journal of Automated Reasoning (JAR), to a- pear.. B. Benhamou and L. Sais. Theoretical s- tudy of symmetries in propositional calculus and application. Eleventh International Conference on Automated Deduction, Saratoga Springs,NY, USA, R. Dechter. From local to global consistency. Articial Intelligence, 55, pages 87{17, 1992.

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